USA-Chile Workshop on Nonlinear Analysis, Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 267–283. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)

Nontrivial solutions for noncooperative elliptic systems at resonance ^{∗}

Elves A. B. Silva

Abstract

In this article we establish the existence of a nonzero solution for variational noncooperative elliptic systems under Dirichlet boundary conditions and a resonant condition at infinity. Situations where the problem is nonresonant and resonant at the origin are considered. The results are based on a version of a critical point theorem for strongly indefinite functionals which are asymptotically quadratic at infinity and do not satisfy any Palais-Smale type condition.

1 Introduction

In this article we consider the existence of a nonzero solution for the variational noncooperative elliptic system

−Δu = au − bv + f(x, u, v) =: F_{u}(x, u, v), in Ω,
−Δv = bu + cv − g(x, u, v) =: −F_{v}(x, u, v), in Ω,  (1.1)
u = v = 0, on ∂Ω ,

where Ω is a bounded domain in ℝ^{N} with smooth boundary ∂Ω, the numbers a, b, c are real parameters and the nonlinearities f, g : Ω̄ × ℝ^{2} → ℝ satisfy f(x, 0) ≡ 0, g(x, 0) ≡ 0. To apply the infinite dimensional Morse theory to the functional associated with (1.1), we assume that F : Ω̄ × ℝ^{2} → ℝ is of class C^{2} and satisfies the growth condition

|D^{2}F (x, z)| ≤ c_{1}|z|^{σ−2} + c_{2}, ∀ z ∈ ℝ^{2}, x ∈ Ω,

for constants c_{1}, c_{2} > 0 and σ > 2 (σ < \frac{2N}{N−2}, if N ≥ 3). In this paper we represent by ∇F (x, z) and D^{2}F (x, z), respectively, the gradient of F and the second derivative of F with respect to the variable z ∈ ℝ^{2}.

∗Mathematics Subject Classifications: 35J65, 58E05.

Key words: Noncooperative systems, Variational Methods, Resonant problem.

\begin{matrix}c \\ ◯ \end{matrix}2001 Southwest Texas State University.

Published January 8, 2001.

partially supported by CNPq/Brazil.

We observe that there exists a vast literature on the use of nonlinear methods to the study of elliptic systems. We refer the readers to the works by Figueiredo-Mitidieri [13], Lazer-Mckenna [20], Silva [30], Costa-Magalhães [10, 11], Felmer-Figueiredo [12], Hulshof-van der Vorst [16], Kryszewski-Szulkin [18] and references therein.

One of our main goals in this article is to illustrate how the ideas introduced in [31] can be applied to handle the problem of existence of a nonzero solution for the system (1.1) under a resonant condition at infinity [19, 26, 11]. More specifically, here we assume

(F_{1}) lim_{|z|→∞} \frac{|(f,g)(x,z)|}{|z|} = 0, for a. e. x ∈ Ω, |(f, g)(x, z)| ≤ A(x)|z| + B(x), for all z ∈ ℝ^{2}, for a. e. x in Ω, where B ∈ L^{1}(Ω) and A ∈ L^{p_{1}}(Ω),

1 < p_{1} < ∞ (p_{1} = N/2, if N ≥ 3)),

We also assume that the associated linear system

−Δu = au − bv, in Ω,
−Δv = bu + cv, in Ω,  (1.2)
u = v = 0, on ∂Ω ,

possesses a nontrivial solution.

In this work the existence of a nonzero solution for (1.1) is based on the relation between the index of the second derivative of the associated functional at zero and at points away from zero and on an appropriate region containing the space of solutions of (1.2). In order to obtain such estimates, we impose conditions on the behavior of the nonlinearity at the origin and at infinity:

(F_{2}) F (x, 0) ≡ 0, ∇F (x, 0) ≡ 0, and D^{2}F (x, 0) = RA_{0},

(F_{3}) lim inf_{|z|→∞} D^{2}F (x, z) > RA_{1}, for a.e. x ∈ Ω, D^{2}F (x, z) ≥ −C(x)I_{2}, for all z in ℝ^{2}, for a.e. x in Ω, where C ∈ L^{p_{2}}(Ω), 1 < p_{2} < ∞ (p_{2} = N/2, if N ≥ 3), I_{2} is the identity 2 × 2 matrix, and

R = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) , A_{i} = ( \begin{matrix}a_{i} & −b_{i} \\ b_{i} & c_{i} \end{matrix} ) , i = 0, 1.

Before stating our basic theorems, we recall some facts about the spectrum of the linear system (1.2) and introduce some notation.

Let 0 < λ_{1} < … ≤ λ_{j} ≤ … be the sequence of eigenvalues of the operator −Δ on H_{0}^{1}(Ω). Denoting by A the anti-symmetric 2 × 2 matrix associated to (1.2), it can be proved (see [10]) that μ is an eigenvalue of the corresponding linear operator on H_{0}^{1}(Ω) × H^{1}(Ω) if, and only if, det(A + μR − λ_{j}I_{2}) = 0. Therefore, the corresponding sequence of eigenvalues of (1.2), {μ_{j}^{±}}, is given by

μ_{j}^{±} = \frac{c − a}{2} ± \sqrt{(\frac{c − a}{2})^{2} + det(λ_{j}I_{2} − A)}.

For a given k ∈ ℕ, we denote by i_{k}(A) the number of negative eigenvalues of the set {μ_{−k}, …, μ_{k}} minus k. Observing that μ_{j}^{(±)} → (±)∞, as j → ∞, we conclude that i_{k}(A) is constant for k sufficiently large. That allows us to define the relative index of A, denoted i(A), by

i(A) = lim_{k→∞} i_{k}(A).

By the same reasoning, we may define the nullity of A, represented by n(A), as the number of eigenvalues of (1.2) which are zero. Note that i(A) is the relative Morse index of the quadratic form associated to (1.2) on H_{0}^{1}(Ω) × H_{0}^{1}(Ω) [18]. Furthermore, the linear system (1.2) has a nontrivial solution if, and only if, n(A) ≠ 0. We now state our first result:

Theorem 1.1 Suppose F ∈ C^{2}(Ω̄ × ℝ^{2}, ℝ) satisfies (F_{1}) − (F_{3}), with n(A) ≠ 0 and n(A_{0}) = 0. Then the system (1.1) possesses a nonzero solution provided i(A_{1}) > i(A_{0}) + 1.

We remark that n(A_{0}) ≠ 0 if, and only if, the origin is a nondegenerate critical point of the associated functional. On that case the problem (1.1) is said to be resonant at the origin. To deal with such possibility we suppose the local condition:

(F_{4}) there exists r > 0 such that F (x, z) ≤ \frac{1}{2}⟨A_{0}z, z⟩, for all |z| ≤ r, for a.e. x ∈ Ω.

The following theorem give us a version of Theorem 1.1 under the resonant condition at the origin.

Theorem 1.2 Suppose F ∈ C^{2}(Ω̄ × ℝ^{2}, ℝ) satisfies (F_{1}) − (F_{3}), with n(A) ≠ 0 and n(A_{0}) ≠ 0. Then the system (1.1) possesses a nonzero solution provided F satisfies (F_{4}) and i(A_{1}) > i(A_{0}) + 1.

Assuming the following version of (F_{4})

(F̂_{4}) there exists r > 0 such that F (x, z) ≥ \frac{1}{2}⟨A_{0}z, z⟩,for all |z| ≤ r, for a.e. x ∈ Ω,

we obtain

Theorem 1.3 Suppose F ∈ C^{2}(Ω̄ × ℝ^{2}, ℝ) satisfies (F_{1})-(F_{3}), with n(A) ≠ 0 and n(A_{0}) ≠ 0. Then the system (1.1) possesses a nonzero solution provided F satisfies (F̂_{4}) and i(A_{1}) > i(A_{0}) + n(A_{0}) + 1.

Theorems 1.1, 1.2 and 1.3 are related to earlier results by Costa-Magalhães [10, 11] and Kryszewski-Szulkin [18]. We note that, under conditions (F_{1}) − (F_{3}), the associated functional may not satisfy any version of the Palais-Smale compactness condition [27, 4] on levels belonging to a very general subset of the real line. This fact does not allow us to apply the abstract results considered in [10, 11, 18]. We should also note that versions of the above theorems can be proved when we assume that lim sup_{|z|→∞} D^{2}F (x, z) is bounded from above by RA_{1} (see Remark 5.8).

As observed earlier, the proofs of Theorems 1.1, 1.2 and 1.3 are based on a critical point theorem, due to the author [31], for strongly indefinite functionals which are asymptotically quadratic at infinity. In this article, we establish a slightly improved version of that result. We believe that the proof given here is more clarifying than the one in [31].

We should observe that the method applied here, and in [31], for establishing critical point theorems when the functional does not satisfy a compactness condition is based on perturbation arguments, a construction associated with the existence of a local linking structure [22, 28, 29], and the estimates established by Lazer-Solimini [21, 33] for the Morse index of a functional at a critical point associated to a given minimax level.

It is worthwhile mentioning that the above method is related to the methods used by Masiello-Pisane [25], Hirano-Li-Wang [15] and Li-Wang [23] to study the scalar problem under strongly resonant conditions at infinity. Note that the conditions (F_{1}) − (F_{3}) include such class of problems and that we are assuming pointwise limits in conditions (F_{1}) and (F_{3}). Finally, we should remark that corresponding results for periodic solutions of asymptotically linear Hamiltonian systems have been derived in [5, 32, 9, 1].

This article has the following structure: in section 2, we state the main critical point theorem. In section 3, after presenting some preliminary results, we prove an abstract theorem that gives us an estimate for the Morse index of the the functional at a critical point belonging to a level c ≠ 0. The main critical point theorem is proved in section 4 by applying the theorem of section 3 and a perturbation argument. Finally, we reserve the section 5 for the proofs of Theorems 1.1, 1.2 and 1.3.

2 A critical point theorem

Let H be a real separable Hilbert space with the inner product ⟨·, ·⟩ and let I : H → ℝ be a functional of class C^{2} having the origin as a critical point. Our goal in this section is to provide sufficient conditions for the existence of a nonzero critical point for the functional when I is of the form

I(u) = \frac{1}{2}⟨Lu, u⟩ + J(u),  (2.1)

where L is a bounded self-adjoint linear operator and J is a functional of class C^{2} satisfying, respectively,

(I_{1}) the number zero is an isolated point of the spectrum of L with finite multiplicity,

(I_{2}) J′ : H → H is compact and lim_{∥u∥→∞} \frac{∥J′(u)∥}{∥u∥} = 0.

Note that the condition (I_{2}) implies that I is asymptotically quadratic at infinity, and the condition (I_{1}) says that the associated quadratic form is degenerate.

Invoking the spectral theory for self-adjoint operators, we may write H = H^{+} ⊕ H^{0} ⊕ H^{−}, where H^{+}, H^{−}, H^{0} are the orthogonal closed subspaces of H on which L is strictly positive definite, strictly negative definite and null, respectively. Furthermore, H^{0} is a nontrivial finite dimensional subspace of H. We also recall that the index of L, denoted ind(L), is the dimension of the subspace H^{−}.

Here, we are interested in the situation where I is strongly indefinite in the sense that both subspaces H^{+} and H^{−} have infinite dimensions. This is indeed the case for the noncooperative elliptic system considered in this article.

In order to apply a Galerking type argument in our setting, we assume the existence of a family of closed subspaces H_{k} = H_{k}^{+} ⊕ H^{0} ⊕ H_{k}^{−} of H such that H_{1}^{±} ⊂ … ⊂ H_{k}^{±} ⊂ … ⊂ H^{±}. If dim(H^{±}) < ∞, we set H_{k}^{±} = H^{±} for every k ∈ ℕ. We also suppose the existence of a basis {e_{j} | j ∈ J ⊂ ℕ} of H such that, for every j ∈ J, there exists k_{j} ∈ ℕ so that {e_{1}, … e_{j}} ⊂ H_{k_{j}}. In the following we denote by I_{k} the restriction of the functional I to the subspace H_{k}.

We now recall the versions of the Palais-Smale condition associated to the family (H_{k}) [3, 22, 28, 29]:

Definition 2.1 (i) Given c ∈ ℝ, we say that (u_{j}) ⊂ H is a (P S)_{c}^{∗}-sequence, for c ∈ ℝ, if I(u_{j}) → c, as j → ∞, and there exists (k_{j}) ⊂ ℕ, with k_{j} → ∞, such that u_{j} ⊂ H_{k_{j}}, for every j ∈ ℕ, and ∥I′_{k_{j}}(u_{j})∥ → 0, as j → ∞. (ii) we say that I ∈ C^{1}(H, ℝ) satisfies the [(P SB)_{c}^{∗}] (P S)_{c}^{∗} condition if every [bounded] (P S)_{c}^{∗}-sequence possesses a convergent subsequence. (iii) If I satisfies [(P SB)_{c}^{∗}] (P S)_{c}^{∗} on every level c ∈ ℝ, we simply say that I satisfies [(P SB)^{∗}] (P S)^{∗} .

It is worthwhile mentioning that we may have dim H_{k} = ∞, for every k ∈ ℕ. Furthermore, when H_{k} = H for k sufficiently large, the Definition 2.1 provides exactly the definitions of the (PS) and the (PSB) conditions.

In this work we also suppose the following local condition [22]:

(I_{3}) there exist ρ > 0 and subspaces X_{k}^{i}, i = 1, 2, of H_{k}, for every k ∈ ℕ, such that H_{k} = X_{k}^{1} ⊕ X_{k}^{2}, dim X_{k}^{1} < ∞, and

(i) I(u) ≥ 0, ∀ u ∈ X_{k}^{2} ∩ B_{ρ}(0),
(ii) I(u) ≤ 0, ∀ u ∈ X_{k}^{1} ∩ B_{ρ}(0).

We observe that the above generalization of the local linking condition has been introduced in [28] (see also [29, 7]). Note that under the condition (I_{3}) the origin may be a degenerate critical point of I and the Morse index of the functional at that point may be infinite.

Given α > 0, we set C_{α} = {u ∈ H : ∥u∥ ≤ (1 + α)∥P_{0}u∥}, where P_{0} denotes the orthogonal projection of H onto ker(L) = H^{0}. The following conditions allow us to compare the Morse index of the functional at the origin with the indexes of L and of D^{2}I(u), for u ∈ C_{α}, with ∥u∥ sufficiently large.

(I_{4}) dim X_{k}^{1} + 1 ≤ dim(H_{k}^{−} ⊕ H^{0}), for every k ∈ ℕ, (I_{5}) there exist α, μ > 0 and M > 0 such that dim X_{k}^{1}+1 < ind(D^{2}I_{k}(u)+μid),

for every u ∈ H_{k} ∩ C_{α}, ∥u∥ ≥ M.

Condition (I_{5}) is slightly weaker than the corresponding condition assumed in [31]. Now, we state our main abstract theorem.

Theorem 2.2 Let H be a real separable Hilbert space. Suppose I ∈ C^{2}(H, ℝ) satisfies (I_{1})−(I_{5}) and (P SB)^{∗}. Then the functional I possesses a critical point other than zero.

The proof of Theorem 2.2 is based on another abstract result which is proved in section 3. In that section we suppose that L is an isomorphism and that H_{k} = H, for every k ∈ ℕ. We also assume that the condition (I_{3}) has some special structure (see Definition 3.2). Under those assumptions, we establish a critical point theorem that relates the Morse index at a nonzero critical point of the functional with the dimension of X_{k}^{1} ≡ X^{1}. The Theorem 2.2 is proved in section 4 by applying the theorem of section 3 to an appropriate perturbation of the functional I_{k}, for k sufficiently large.

3 A version of Theorem 2.2

In this section we suppose that H_{k} = H for every k ∈ ℕ. Under this hypothesis and supposing that L, given by (2.1), is an isomorphism, we are able to provide an estimate for the Morse index of the functional at a critical point associated to a level c ≠ 0.

In this article we represent by K(I) the set of critical points of I. Given c ∈ ℝ, we set I^{c} = {u ∈ H : I(u) ≤ c}, K_{c}(I) = {u ∈ H : I(u) = c, I′(u) = 0}. We also set K_{a}^{b}(I) = ∪_{c=a}^{b}K_{c}(I), for every a < b ∈ ℝ .

Denoting by S the family of continuous map Φ ∈ C([0, 1] × H, H) such that Φ(0, ·) = id and considering the subsets S and Q of H, we say that S and ∂Q link if Φ(t, Q)∩S ≠ ∅, for every 0 ≤ t ≤ 1, whenever Φ ∈ S and Φ([0, 1]×∂Q)∩S = ∅ [6, 27]. The following characterization of the link was introduced in [28] (see also [29, 7]):

Definition 3.1 Given I ∈ C^{1}(H, ℝ), we say that the link between S and ∂Q is of deformation type (with respect to I) if there exist γ′ ≤ γ and Φ ∈ S such that

(L_{1}) Φ(t, ∂Q) ∩ S = ∅, for all t ∈ [0, 1],

(L_{2}) Φ(1, ∂Q) ⊂ I^{γ′},

(L_{3}) I(u) > γ, for all u ∈ S.

We now introduce the corresponding notion when the functional I satisfies the local condition (I_{3}).

Definition 3.2 We say that the condition (I_{3}) is a local linking of deformation type (with respect to I) if there exist 0 < c_{1} < c, 0 < d < r < ρ, a homeomorphism ψ : H → H and a continuous map η : [0, 1] × ∂B_{ρ}(0) ∩ X^{1} → H satisfying

(d_{1}) ψ(u) = u, for all u ∈ (H \ B_{r}(0)) ∪ X^{1},

(d_{2}) I(ψ(u)) ≥ c > 0, for all u ∂B_{d}(0) ∩ X^{2},

(d_{3}) η(0, u) = u, for all u ∈ ∂B_{ρ}(0) ∩ X^{1},

(d_{4}) I(η(t, u)) ≤ −ct, for all (t, u) ∈ [0, 1] × ∂B_{ρ}(0) ∩ X^{1},

(d_{5}) r < ∥η(t, u)∥ < 2ρ, for all (t, u) ∈ [0, 1] × ∂B_{ρ}(0) ∩ X^{1},

(d_{6}) I(u) > −c_{1}, for all u ∈ B_{r}(0).

Remark 3.3 If the functional I has the origin as an isolated critical point and satisfies the (P SB) condition, we may apply the local deformation lemmas proved in [28, 29] to conclude that the condition (I_{3}) is a local linking of deformation type.

Representing by m(I, u) the Morse index of I ∈ C^{2}(H, ℝ) at the critical point u ∈ H, we may state the deformation lemma:

Lemma 3.4 Let ϕ : A → I^{b} be a continuous map, with A ⊂ ℝ^{n} compact. Suppose I ∈ C^{2}(H, ℝ) satisfies (P S)_{c} for every c ∈ [a, b]. Assume that K_{a}^{b}(I) = {u_{j}}_{j=1}^{m}, with u_{j} non-degenerate and m(I, u_{j}) > n, for every 1 ≤ j ≤ m. Then, there exists a continuous map τ : [0, 1] × A → I^{b} such that

(τ_{1}) τ(0, u) = ϕ(u), for all u ∈ A,

(τ_{2}) τ(t, u) = ϕ(u), for all t ∈ [0, 1], if ϕ(u) ∈ I^{a},

(τ_{3}) τ(1, A) ⊂ I^{a}.

Remark 3.5 The Lemma 3.4 is proved by applying the second deformation lemma [8] combined with a local deformation based on the Morse Lemma for nondegenerate critical points (see Lemma 2.9 in [31]). As observed in [31] (see also Lemma 3.10), the above result and the perturbation argument due to Marino-Prodi [24, 21] provide some useful estimates for the Morse index at a nontrivial critical point for functionals satisfying the local linking condition (I_{3}).

Assuming the version of the condition (I_{1})

(Î_{1}) L is an isomorphism,

we state the main result of the section:

Theorem 3.6 Let H be a real Hilbert space. Suppose I ∈ C^{2}(H, ℝ) satisfies (Î_{1}), (I_{2}), (I_{3}) and (I_{4}), with H_{k} = H, for every k ∈ ℕ, and (I_{3}) a local linking of deformation type. Then I possesses a critical point u ∈ H such that either I(u) ≤ −c and m(I, u) ≤ dim X^{1}, or I(u) ≥ c and m(I, u) ≤ dim X^{1} + 1, where c > 0 is given by Definition 3.2.

Remark 3.7 (i) The conditions (Î_{1}) and (I_{2}) imply that K(I) is compact. Consequently, under those hypotheses, the functional satisfies the Palais-Smale condition. (ii) If X^{1} = {0}, we may apply the Mountain Pass Theorem [2] and the corresponding Morse index estimate [8] to obtain a critical point u of I such that I(u) ≥ c and m(I, u) ≥ 1. (iii) If dim X^{2} = 1, we use (I_{4}) and (Î_{1}) to conclude that H = H^{−} and that I has a critical point u such that I(u) = max{I(v) | v ∈ H} ≥ c. Clearly, m(I, u) ≤ dim X^{1} + 1.

Proof of Theorem 3.6: Supposing that m(I, u) > dim X^{1} for every critical point u ∈ H such that I(u) ≤ −c, we prove the Theorem 3.6 by verifying that I has a critical point u_{0} such that I(u_{0}) ≥ c and m(I, u_{0}) ≤ dim X^{1} + 1. That verification is lengthy so we will sketch it first. We start by showing the existence of a functional which is quadratic away from zero and has the same set of critical points than I. Furthermore, that functional will be equal to I on a ball containing K(I). Using a perturbation argument due to Marino-Prodi and Lemma 3.4, we verify the existence of a linking of deformation type and of a critical value b ≥ c. We conclude the proof of Theorem 3.6 by applying the Morse index estimates for the minimax critical points provided by Lazer-Solimini [21, 33].

By Remark 3.7, we may assume without loss of generality that X^{1} ≠ {0} and dim(X^{2}) ≥ 2. Considering

(Î_{2}) I is bounded on bounded sets, and there exists R_{0} > 0 such that J(u) = 0, for every u ∈ H such that ∥u∥ ≥ R_{0},

The next lemma provides the first step for the proof of Theorem 3.6 (see [31]).

Lemma 3.8 Suppose I ∈ C^{2}(H, ℝ) is given by (2.1) and satisfies (Î_{1}) and (I_{2})−(I_{4}), with (I_{3}) a local linking of deformation type. Let R_{1} > 2ρ, with ρ > 0 given by (I_{3}), be such that K(I) ⊂ int(B_{R_{1}}(0)). Then there exists I_{1} ∈ C^{2}(H, ℝ) satisfying (P S), (Î_{1}), (Î_{2}), (I_{3}), (I_{4}), and

I_{1}(u) = I(u), ∀ u ∈ H, ∥u∥ ≤ R_{1}.  (3.1)

Furthermore, K(I_{1}) = K(I).

Remark 3.9 (i) Since R_{1} > 2ρ, (I_{3}) is a local linking of deformation type with respect to I_{1}. Moreover, the constants {c_{1}, c, d, r, ρ}, given in Definition 3.2, are the same for I and I_{1}. (ii) By lemma 3.8, to prove the Theorem 3.6 it suffices to verify that I_{1} has a critical point u_{0} such that I_{1}(u_{0}) ≥ c and m(I_{1}, u_{0}) ≤ dim X^{1} + 1.

Let I_{1} be the functional given by Lemma 3.8. Considering r > 0 given by Definition 3.2 and R_{0} given in (Î_{2}), we fix R > R_{0} such that

\{u = v_{1} + v_{2}, v_{i} ∈ X^{i}, i = 1, 2, : \frac{∥v_{i}∥}{ρ} + \frac{∥v_{2}∥}{R} = 1\} ∩ B_{r}(0) = ∅.  (3.2)

We now take e ∈ H^{−} ∩ X^{2}, ∥e∥ = 1, and we define Q = Q(R) ⊂ ℝe ⊕ X^{1} by

Q = \{u = se + v : v ∈ X^{1}s ≥ 0, \frac{s}{R} + \frac{∥v∥}{ρ} ≤ 1\}.

Writing u = u^{+} + u^{−}, u^{i} ∈ H^{i}, i = +, −, and considering that I_{1} satisfies (Î_{1}) and (Î_{2}), we find a < −c such that

I(u) > a, ∀ u = u^{+} + u^{−}, ∥u^{−}∥ ≤ R.  (3.3)

Lemma 3.10 Let a ∈ ℝ be given by (3.3). Suppose m(I, u) > dim X^{1} for every u ∈ K(I) ∩ I^{−c}. Then there exists a continuous map η_{1} : [0, 1] × ∂B_{ρ}(0) → H satisfying

(η_{1}) η_{1}(0, u) = u, for all u ∈ ∂B_{ρ}(0) ∩ X^{1},

(η_{2}) I_{1}(η_{1}(t, u)) ≤ 0, for all (t, u) ∈ [0, 1] × ∂B_{ρ}(0) ∩ X^{1},

(η_{3}) ∥η_{1}(t, u)∥ > r, for all (t, u) ∈ [0, 1] × ∂B_{ρ}(0) ∩ X^{1},

(η_{4}) η_{1}(1, ∂B_{ρ}(0) ∩ X^{1}) ⊂ I_{1}^{a},

(η_{5}) ∥η_{1}^{−}(1, u)∥ ≥ R, for all u ∈ ∂B_{ρ}(0) ∩ X^{1}.

Proof: First, we claim that there exists b ∈ (c_{1}, c) such that m(I_{1}, u) > dim X^{1}, for every u ∈ K(I_{1}) ∩ I_{1}^{−b}. Effectively, assuming otherwise, by Lemma 3.8, we find (u_{m}) ⊂ K(I) such that I(u_{m}) → −c, as m → ∞, and m(I, u_{m}) ≤ dim X^{1}, for every m ∈ ℕ. Since K(I) is compact, we may suppose that u_{m} → u ∈ K(I), as m → ∞. Hence, I(u) = −c and m(I, u) ≤ dim X^{1}. This prove the claim.

Now, considering 0 < ϵ < (c − b)/2, we use the condition (I_{2}), K(I_{1}) = K(I) ⊂ int B_{R_{1}}(0) , (3.1) and the perturbation method introduced by Marino-Prodi [24] to find I ∈ C^{2}(H, ℝ) such that K(I_{2})∩I_{2}^{−c+ϵ} is a finite set possessing only non-degenerate critical points,

∥I_{2}(u) − I_{1}(u)∥ < ϵ, ∀ u ∈ H.  (3.4)

and

m(I_{2}, u) > dim X^{1}, ∀ u ∈ K(I_{2}) ∩ I_{2}^{−c+ϵ}.  (3.5)

Taking η given by Definition 3.2, from (d_{4}) and (3.4), we have that η(1, .) : ∂B_{ρ}(0) ∩ X^{1} → I_{2}^{−c+ϵ}. Hence, invoking (3.5) and Lemma 3.4, we find a continuous map τ : [0, 1] × ∂B_{ρ}(0) ∩ X^{1} → I_{2}^{−c+ϵ} satisfying

τ(0, u) = η(1, u), ∀ u ∈ ∂B_{ρ}(0) ∩ X^{1},  (3.6)

τ(1, ∂B_{ρ}(0) ∩ X^{1}) ⊂ I_{2}^{a−ϵ}.  (3.7)

Now, we define η_{1} : [0, 1] × ∂B_{ρ}(0) ∩ X^{1} → H by

η_{1}(t, u) = \{ \begin{matrix}η(2t, u), & 0 ≤ t ≤ \frac{1}{2}, u ∈ ∂B_{ρ}(0) ∩ X^{1}, \\ τ(2t − 1, η(1, u)), & \frac{1}{2} < t ≤ 1, u ∈ ∂B_{ρ}(0) ∩ X^{1}. \end{matrix}

By (3.6), η_{1} is a well defined continuous map. We shall verify that η_{1} satisfies the conditions (η_{1}) − (η_{5}). First, we note that (η_{1}) is a direct consequence of the definition of η_{1} and (d_{3}). Considering that −c + 2ϵ < −c_{1}, from (3.4), we get that

τ([\frac{1}{2}, 1] × ∂B_{ρ}(0) ∩ X^{1}) ⊂ int(I_{1}^{−c_{1}}).  (3.8)

This fact and (d_{4}) show that η_{1} satisfies (η_{2}). The relation (3.8), (d_{5}) and (d_{6}) imply that (η_{3}) holds. The condition (η_{4}) is a consequence of (3.4) and (3.7). Finally, we observe that (η_{5}) is implied by (η_{4}) and (3.3). The Lemma 3.10 is proved. ♢

Invoking (η_{5}), R > R_{0} and (Î_{2}), we obtain

I_{1}(sη_{1}^{+}(1, u) + η_{1}^{−}(1, u)) ≤ 0, ∀ u ∈ ∂B_{ρ}(0) ∩ X^{1}, 0 ≤ s ≤ 1.

Using the above relation, (η_{1}) − (η_{5}) and the fact that dim(X^{2}) ≥ 2, by [28] (see also Lemma 1.25 in [29]), we have

Proposition 3.11 There exists Φ ∈ S satisfying

(L̂_{1}) Φ([0, 1] × ∂Q) ⊂ {u ∈ H | ∥u∥ ≥ r} ∪ X^{1},

(L̂_{2}) Φ(1, ∂Q) ⊂ I_{1}^{0}.

Conclusion of the proof of Theorem 3.6: From (3.2) and (d_{1}), we have that S = ψ(∂B_{d}(0) ∩ X^{2}) and ψ(∂Q) = ∂Q since 0 < d < r and ψ is a homeomorphism. We claim that this link is of deformation type with respect to I_{1}. Effectively, taking Φ ∈ S given by Proposition 3.11, we see easily that (L_{2}) and (L_{3}) hold with γ ≥ c > 0 = γ′. Moreover, the condition (L_{1}) is a consequence of (d_{1}), (L̂_{1}) and d < r. The claim is proved.

As I_{1} satisfies (P S), we may invoke [28, 29] to conclude that I_{1} has a critical value b ≥ c > 0 given by

b = inf_{Φ∈Γ} max_{u∈Q} I_{1}(Φ(1, u)),  (3.9)

where

Γ = {Φ ∈ S : Φ satisfies (L_{1}) and (L_{2})}.  (3.10)

Finally, considering that Q ⊂ ℝe ⊕ X^{1} we may apply the Morse index estimates for minimax critical points given by [21, 33] to conclude that I_{1} has a critical point u_{0} ∈ H such that I_{1}(u_{0}) = b ≥ c and m(I_{1}, u_{0}) ≤ dim X^{1} + 1. That concludes the proof of Theorem 3.6. ♢

4 Proof of Theorem 2.2

Arguing by contradiction, we suppose that u = 0 is the only critical point of I in H. First, we note that given 0 < r_{1} < R_{1}, from (I_{1}), (I_{2}) and (P SB)^{∗}, there exist δ > 0 and k_{0} ∈ ℕ such that

∥I′_{k}(u)∥ ≥ δ > 0, ∀ r_{1} ≤ ∥u∥ ≤ R_{1}, k ≥ k_{0}.  (4.1)

The next lemma is a direct consequence of the local deformations lemmas proved in [28, 29] (see also Lemma 3.8 in [31]).

Lemma 4.1 There exists k_{1} ∈ ℕ such that (I_{3}) is a local linking of deformation type with respect to I_{k}, for every k ≥ k_{1}. Furthermore, the constants {c, c_{1}, r, d, ρ} appearing in Definition 3.2 are independent of k ≥ k_{1}.

Consider c, ρ > 0 given by Lemma 4.1 and take 0 < β < c. Since I(0) = 0, there exists 0 < r_{1} < ρ so that

|I(u)| ≤ β, ∀ u ∈ B_{r_{1}}(0).  (4.2)

Fixing R > 2ρ and using (I_{1}) − (I_{2}) and (P SB)^{∗}, we find k_{2} ≥ k_{1} and δ̂ > 0 such that, for every k ≥ k_{2},

∥I′_{k}(u)∥ ≥ δ̂ > 0, ∀ r_{1} ≤ ∥u^{0}∥ ≤ R + 1.  (4.3)

Now, taking χ : ℝ → [0, 1] of class C^{∞} such that χ(s) = 0, if s ≤ 0, and χ(s) = 1, if s ≥ 1, we set χ_{R}(s) = χ(s − R) and define I_{k,ϵ} ∈ C^{2}(H, ℝ) by

I_{k,ϵ} = I_{k}(u) − \frac{ϵ}{2}χ_{R}(∥u^{0}∥)∥u^{0}∥^{2}, ∀ u ∈ H,  (4.4)

for k ≥ k_{2} and ϵ > 0. We take M̂ > {M, (R + 1)/(α + 1)}, M, α > 0 given by (I_{5}), and use (4.4), (I_{1}) − (I_{2}) and (P SB)^{∗} to obtain k ≥ k_{2} and ϵ > 0 so that

M̂ < ∥u∥ < (1 + α)∥u^{0}∥, ∀ u ∈ K(I_{k,ϵ}) \ B_{r_{1}}(0),  (4.5)

Setting Î = I_{k,ϵ}, from (4.3), (4.5) and (I_{5}), we get

m(Î, u) > dim X_{k}^{1} + 1, ∀ u ∈ K(Î) \ B_{r_{1}}(0).  (4.6)

On the other hand, by (4.4) and Lemma 4.1, we obtain that Î satisfies (Î_{1}), (I_{2}) − (I_{4}), with (I_{3}) a linking of deformation type with respect to Î. Thus, by Theorem 3.6, Î possesses a critical point u_{0} ∈ H such that |Î(u_{0})| ≥ c > 0, c given by Lemma 4.1, and m(Î, u_{0}) ≤ dim X_{k}^{1} + 1. Noting that Î(u) = I(u), for every u ∈ B_{r_{1}}(0), we obtain a contradiction with β < c, (4.2) and (4.6). The proof of Theorem 2.2 is complete. ♢

We finish this section by presenting a version of Theorem 2.2 when (H_{k}) satisfies H_{k} = H, for every k ∈ ℕ.

Theorem 4.2 Let H be a real Hilbert space. Suppose I ∈ C^{2}(H, ℝ) satisfies (I_{1})−(I_{5}), with H_{k} = H, for every k ∈ ℕ. Assume that the origin is an isolated critical point of I. Then there exists c > 0 such that I possesses a critical point u ∈ H satisfying either I(u) ≤ −c and m(I, u) ≤ dim X^{1}, or I(u) ≥ c and m(I, u) ≤ dim X^{1} + 1.

Remark 4.3 The Theorem 4.2 generalizes the Theorem 2.18 in [31].

5 Proofs of Theorems 1.1, 1.2 and 1.3

We start by recalling the variational structure associated to the problem (1.1). Consider the Hilbert space H = H_{0}^{1}(Ω)×H_{0}^{1}(Ω) endowed with the inner product

⟨z, ξ⟩ = ∫_{Ω}⟨∇z, ∇ξ⟩ dx, ∀ z, ξ ∈ H,

where ∇z = (∇u, ∇v), for z = (u, v) ∈ H. Denoting by I : H → ℝ the functional associated to (1.1), we may write

I(z) = Q_{A}(z) + J(z), ∀ z ∈ H,  (5.1)

with

Q_{A}(z) = \frac{1}{2}⟨L_{A}z, z⟩ = \frac{1}{2} ∫_{Ω} (⟨R∇z, ∇z⟩ − ⟨RAz, z⟩) dx,  (5.2)

J(z) = ∫_{Ω} G(x, z) dx,  (5.3)

and G(x, z) = \frac{1}{2}⟨RAz, z⟩ − F (x, z). We note that a standard argument [27] shows that I ∈ C^{2}(H, ℝ) and that the critical points of I are solutions of (1.1). Furthermore, we have that the self-adjoint linear operator L_{A} : H → H, given by (5.2), satisfies the condition (I_{1}). In the following, we denote by H^{+}(A), H^{0}(A) and H^{−}(A) the spectral decomposition of H associated to L_{A}. We also set T_{A} = L_{A} − L_{0}, where L_{0} is the linear operator associated to the null matrix.

For every j ∈ ℕ, we set e_{j} = (φ_{j}, 0) and e_{−j} = (0, φ_{j}), where φ_{j} is the eigenvector associated to the eigenvalue λ_{j} of the operator −Δ on H_{0}^{1}(Ω). We note that H^{+}(0) = \overline{span\{e_{j} | j ∈ ℕ\}} and H^{−}(0) = \overline{span\{e_{−j} | j ∈ ℕ\}}, where H^{+}(0) and H^{−}(0) are the subspaces given by the spectral decomposition of H associated to the linear operator L_{0}.

In order to apply the Theorem 2.2 to our problem, we consider the family of closed subspaces (H_{k}) of H defined by H_{k} = H_{k}^{−}(0) ⊕ H^{+}(0), where H_{k}^{−}(0) = span{e_{−1}, …, e_{−k}}, for every k ∈ ℕ. The following lemma is proved in [10, 11].

Lemma 5.1 Let A be an anti-symmetric 2×2 matrix. Then the linear operator T_{A} = L_{A} − L_{0} is compact. Furthermore, for every k ∈ ℕ, H_{k} is an invariant subspace of T_{A}.

Remark 5.2 (i) As a consequence of Lemma 5.1, we have that H_{k} = H^{+}(A)⊕ H^{0}(A) ⊕ H_{k}^{−}(A), for k sufficiently large, where H_{k}^{−}(A) is a finite subspace of H^{−}(A). (ii) Considering the definitions given in the introduction, we obtain that n(A) = dim H^{0}(A) and i(A) = i_{k}(L_{A}) − i_{k}(L_{0}) = dim(H_{k}^{−}(A)) − k, for k sufficiently large, where i_{k}(L) denotes the index of the operator L restricted to H_{k}.

Taking X^{1} = H^{−}(A_{0}) and X^{2} = H^{0}(A_{0}) ⊕ H^{+}(A_{0}), the proof of the next lemma can be based on the argument used in [28, 29] (see also [31]).

Lemma 5.3 Suppose F ∈ C^{2}(Ω̄ × ℝ^{2}, ℝ) satisfies (F_{2}), with (F_{4}) holding when n(A_{0}) ≠ 0. Then there exists ρ > 0 such that

(i) I(u) ≥ 0, for all u ∈ X^{2} ∩ B_{ρ}(0), (ii) I(u) ≤ 0, for all u ∈ X^{1} ∩ B_{ρ}(0).

Setting X_{k}^{i} = X^{i} ∩ H_{k}, i = 1, 2, for every k ∈ ℕ, as a direct consequence of Lemma 5.3, we obtain

Corollary 5.4 Suppose F ∈ C^{2}(Ω̄ × ℝ^{2}, ℝ) satisfies (F_{2}), with (F_{4}) holding when n(A_{0}) ≠ 0. Then the functional I satisfies (I_{3}).

We omit the proof of the next result since it is similar to the proof of Lemma 5.6 (see also [14] and references therein),

Lemma 5.5 Suppose F ∈ C^{2}(Ω̄ × ℝ^{2}, ℝ) satisfies (F_{1}). Then the functional J given by (5.3) satisfies the condition (I_{2}).

The next lemma allows us to handle the pointwise limit assumed in condition (F_{3}).

Lemma 5.6 Suppose F ∈ C^{2}(Ω̄ × ℝ^{2}, ℝ) satisfies (F_{3}). Then, given β > 0, there exists M > 0 and α > 0 such that

∫_{Ω} (⟨D^{2}F (x, z)w, w⟩ − ⟨RA_{1}w, w⟩) dx ≥ −β, ∀ w ∈ ∂B_{1}(0),  (5.4)

for every z ∈ C_{α}, ∥z∥ > M.

Proof: Arguing by contradiction, we suppose that there exist sequences (z_{n}) ⊂ H and (w_{n}) ⊂ ∂B_{1}(0) satisfying ∥z_{n}∥ → ∞, \frac{∥z_{n}^{0}∥}{∥z_{n}∥} → 1, as n → ∞, and

∫_{Ω} g_{n}(x) dx ≤ −β, ∀ n ∈ ℕ,  (5.5)

where g_{n}(x) = ⟨D^{2}F (x, z_{n}(x))w_{n}(x), w_{n}(x)⟩ − ⟨RA_{1}w_{n}(x), w_{n}(x)⟩, for x ∈ Ω. Taking v_{n} = z_{n}/∥z_{n}∥ in ∂B_{1}(0) and using that H^{0} = H^{0}(A) is finite dimensional, we may assume that v_{n}(x) → v(x) ∈ ∂B_{1}(0) ∩ H^{0}. Hence, by the unique continuation property, we must have

|z_{n}(x)| → ∞, for a. e. x ∈ Ω.  (5.6)

Considering N > 2 and invoking the Sobolev Embedding Theorem, we may assume that there exists w ∈ H such that

w_{n} ⇀ w, weakly in H,
w_{n}(x) → w(x), for a. e. x ∈ Ω,  (5.7)
|w_{n}(x)| ≤ h_{q}(x) ∈ L^{q}(Ω), 1 ≤ q < 2N/(N − 2), for a. e. x ∈ Ω.

Observing that (|w_{n}(x)|^{2}) ⊂ L^{N/(N−2)}(Ω) is bounded and that w_{n}(x) → w(x), a. e. on Ω, we may also suppose (see [17]) that |w_{n}(x)|^{2} ⇀ |w(x)|^{2}, weakly in L^{N/(N−2)}(Ω). Thus, since C(x) ∈ L^{\frac{N}{2}} (Ω), we get

lim_{n→∞} ∫_{Ω} C(x)|w_{n}(x)|^{2} dx = ∫_{Ω} C(x)|w(x)|^{2} dx.  (5.8)

Now, we use (5.7) and (F_{3}) one more time to find h ∈ L^{1}(Ω) such that

g_{n}(x) − C(x)|w_{n}(x)|^{2} ≥ h(x), for a. e. x ∈ Ω.

Hence, invoking (5.6), (5.7), (F_{3}), the above inequality and Fatou’s Lemma, we have

lim inf_{n→∞} ∫_{Ω} (g_{n}(x) − C(x)|w_{n}(x)|^{2}) dx ≥ − ∫_{Ω} C(x)|w(x)|^{2} dx.

But, this last relation and (5.8) contradict (5.5). The Lemma 5.6 is proved. ♢

Proposition 5.7 Suppose F ∈ C^{2}(Ω̄ × ℝ^{2}, ℝ) satisfies (F_{2}) − (F_{3}), with (F_{4}) holding when n(A_{0}) ≠ 0. If i(A_{1}) > i(A_{0}) + 1, then the functional I given by (5.1) satisfies the conditions (I_{4}) and (I_{5}).

Proof: First, we invoke the Lemma 5.1 and the Remark 5.2 to find k_{1} ∈ ℕ such that, for every k ≥ k_{1}, we have

i_{k}(L_{A}) = i(A) + k = dim(H_{k}^{−}(A)),
i_{k}(L_{A_{1}}) = i(A_{1}) + k = dim(H_{k}^{−}(A_{1})),
i_{k}(L_{A_{0}}) = i(A_{0}) + k = dim(X_{k}^{1}).

(5.9)

Observing that the conditions (F_{1}) and (F_{3}) imply that L_{A} ≤ L_{A_{1}}, from i(A_{1}) > i(A_{0}) + 1 and (5.9), we get that dim(H_{k}^{−}(A)) ≥ dim(X_{k}^{1}) + 1, for k ≥ k_{1}. Thus, I satisfies the condition (I_{4}).

We now verify that the condition (I_{5}) is satisfied by I: applying the Lemma 5.1 one more time and taking k_{1} larger if necessary, we may suppose that

∫_{Ω}⟨RA_{1}w, w⟩ dx ≤ \frac{1}{2}∥w∥^{2}, ∀ w ∈ H_{k_{1}}^{⊥} ∩ H.  (5.10)

We now consider the subspace V = H_{k_{1}}^{−}(A_{1}) of H_{k_{1}} and take 0 < β < 1/4 so that

⟨L_{A_{1}}v, v⟩ ≤ −2β∥v∥^{2}, ∀ v ∈ V.  (5.11)

Applying Lemma 5.6, we find M > 0, α > 0 such that (5.4) holds for β given above.

Given k > k_{1}, we set Y_{k} = span{e_{−(k_{1}+1)}, …, e_{−k}} and Ĥ_{k} = V ⊕ Y_{k}. Taking w ∈ Ĥ_{k} ∩ ∂B_{1}(0), we write w = v + y, with v ∈ V, y ∈ Y_{k}, and we use (5.4), (5.10) and (5.11), to obtain

⟨D^{2}I_{k}(z)w, w⟩ ≤ −β∥v∥^{2} − (\frac{1}{2} − 2β) ∥y∥^{2},

for every z ∈ C_{α} with ∥z∥ > M. Since, by (5.9), dim(Ĥ_{k}) = dim(V) + k − k_{1} = i(A_{1}) + k > i(A_{0}) + 1 + k = dim X_{k}^{1} + 1, we conclude that I satisfies (I_{5}) for every 0 < μ < \{−β, \frac{1}{2} − 2β\}. That concludes the proof of Proposition 5.7. ♢

Proofs of Theorems 1.1 and 1.2: As observed above the functional I given by (5.1) satisfies (I_{1}). The Corollary 5.4, the Lemma 5.5 and the Proposition 5.6 imply that I satisfies (I_{2}) − (I_{5}). Since H_{k} is an invariant subspace of L_{A} and I satisfies (I_{1}) and (I_{2}), we also have that the (P SB)^{∗} condition is satisfied by I. Invoking the Theorem 2.2, we obtain that I has a critical point other than zero. That concludes the proofs of Theorems 1.1 and 1.2. ♢

Proof of Theorem 1.3: We just observe that under the hypotheses (F̂_{4}) and i(A_{1}) > i(A_{0}) + n(A_{0}) + 1, we may prove the corresponding versions of Lemma 5.3 and Proposition 5.7 by taking X^{1} = H^{−}(A_{0}) ⊕ H^{0}(A_{0}) and X^{2} = H^{+}(A_{0}), respectively. ♢

Remark 5.8 We note that versions of the Theorems 1.1, 1.2 and 1.3 can be proved when we have lim sup_{|z|→∞} D^{2}F (x, z) bounded from above by RA_{1}. On that case, the results should be given in function of the relative numbers of negative eigenvalues of the problem (1.2) associated to the matrices A_{0} and A_{1}.

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Elves A. B. Silva Dpto. Matemática Universidade de Brasília 70910 Brasilia, DF Brazil e-mail address: elves@mat.unb.br